Hi! I want to learn a course of "general relativity". For this, I've realized that I have to master the differential geometry. So, I've chosen Lee's book called " introduction to smooth manifolds". In the appendix of the book, some required knowledege of integrations on an euclidean space are listed. And I found a definition of integral over a bounded set. Suddenly, a question has passed through my brain. I've also tried to learn an integration theory from fleming's book " functions of several variables". But, it seems that the book uses a different definition from lee's book. For example, fleming's integrates a function over a bounded measurable set. On the other hand, lee's book doesn't care about measurable set. the lee's book integrates a function over a just bouded set. This is the point where my problems occur. I think the definitions of integral over a bouded set must be equivalent. But as you can see, they are different. I can't understand why authors don't use a common definition of integration over a bouded set. Do I have to follow the each different definition of a integration of different books?? Then, I believe that it would be difficult to exchange common ideas when talking about something of math. Why do some authurs use one definition of integration and others use others?? Do you think that all following books; munkres's "Analysis on manifolds", spivak's "calculus on manifolds", fleming's "functions of several variables", and so on, use the equivalent definitions of integration over a bounded set?? How do you think about my problems? What is the wrong parts that I've mentioned above?? Please, notice about them for me... This is very serious problem for me.